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 Homework Statement:

The problem is two parts:
i) Let ##X## be a finite ##G##set , and let ##H \le G## act transitively on ##X##. Then ##G = HG_x## for each ##x \in X##.
ii) Show that the Frattini argument follows from i).
 Relevant Equations:

Relevant equations:
Frattini Argument: Let ##K## be a normal subgroup of a finite group ##G##. If ##P## is a Sylow ##p##subgroup of ##K## (for some prime ##p##), then $$G = KN_G(P).$$
##X## is a finite ##G##set means that ##G## acts on ##X## and both ##X## and ##G## are finite.
##G_x = \lbrace g \in G : g\cdot x = x \rbrace##
Attempt at solution:
Proof of i): Let ##x \in X##. Its clear ##G \supseteq HG_x##. Let ##g \in G##.Then there is ##y \in X## such that ##g \cdot x = y##. Since ##H## acts transitively on ##X##, there is ##h \in H## such that ##h \cdot x = y##. So, ##g \cdot x = h \cdot x##. This gives $$(h^{1}g)\cdot x = x$$ Hence, ##g = h(h^{1}g) \in HG_x## and we can conclude ##G = HG_x##. []
For ii), let ##X = \lbrace gPg^{1} : g \in G\rbrace##. Then ##G## acts on ##X## by conjugation and ##N_G(P)## is the stabilizer of ##P##. But I'm not sure if ##K## acts transitively on ##X##. I know I haven't used the fact that ##K## is a normal subgroup of ##G##. Can I have a hint on how to solve ii), please?
Proof of i): Let ##x \in X##. Its clear ##G \supseteq HG_x##. Let ##g \in G##.Then there is ##y \in X## such that ##g \cdot x = y##. Since ##H## acts transitively on ##X##, there is ##h \in H## such that ##h \cdot x = y##. So, ##g \cdot x = h \cdot x##. This gives $$(h^{1}g)\cdot x = x$$ Hence, ##g = h(h^{1}g) \in HG_x## and we can conclude ##G = HG_x##. []
For ii), let ##X = \lbrace gPg^{1} : g \in G\rbrace##. Then ##G## acts on ##X## by conjugation and ##N_G(P)## is the stabilizer of ##P##. But I'm not sure if ##K## acts transitively on ##X##. I know I haven't used the fact that ##K## is a normal subgroup of ##G##. Can I have a hint on how to solve ii), please?